U.S. patent number 6,636,045 [Application Number 10/072,173] was granted by the patent office on 2003-10-21 for method of determining formation anisotropy in deviated wells using separation of induction mode.
This patent grant is currently assigned to Baker Hughes Incorporated. Invention is credited to Mikhail Epov, Michael B. Rabinovich, Leonty A. Tabarovsky.
United States Patent |
6,636,045 |
Tabarovsky , et al. |
October 21, 2003 |
Method of determining formation anisotropy in deviated wells using
separation of induction mode
Abstract
Measurements are made with a multicomponent induction logging
tool in earth formations in a borehole inclined to earth
formations. A combination of principal component measurements is
used to determine the horizontal resistivity of the earth
formations. The determined horizontal resistivities are used in a
model for inversion of other components of the data to obtain the
vertical formations resistivities. When multifrequency measurements
are available, frequency focusing is used.
Inventors: |
Tabarovsky; Leonty A. (Houston,
TX), Epov; Mikhail (Novosibirsk, RU), Rabinovich;
Michael B. (Houston, TX) |
Assignee: |
Baker Hughes Incorporated
(Houston, TX)
|
Family
ID: |
26753071 |
Appl.
No.: |
10/072,173 |
Filed: |
February 7, 2002 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
Issue Date |
|
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825104 |
Apr 3, 2001 |
|
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Current U.S.
Class: |
324/343;
702/7 |
Current CPC
Class: |
G01V
3/38 (20130101); G01V 3/28 (20130101) |
Current International
Class: |
G01V
3/10 (20060101); G01V 3/18 (20060101); G01V
003/18 () |
Field of
Search: |
;324/338-343
;702/6,7,9 |
References Cited
[Referenced By]
U.S. Patent Documents
Other References
J H. Moran et al.; Effects of formation anisotropy on
resistivity-logging measurments, Geophysics, vol. 44, No. 7, Jul.
1979, pp. 1266-1286, 21 Figures, 4 Tables..
|
Primary Examiner: Le; N.
Assistant Examiner: Aurora; Reena
Attorney, Agent or Firm: Madan, Mossman & Sriram,
P.C.
Parent Case Text
CROSS-REFERENCES TO RELATED APPLICATIONS
This application is a continuation-in-part of U.S. patent
application Ser. No. 09/825,104 filed on Apr. 3, 2001. It also
claims priority from U.S. Provisional Patent Application Ser. No.
60/312,655 filed on Aug. 15, 2001. The contents of both documents
are fully incorporated herein by reference.
Claims
What is claimed is:
1. A method of lagging a subsurface formation comprising a
plurality a layers each having a horizontal conductivity and a
vertical conductivity, the method comprising: (a) conveying an
electromagnetic logging tool into a borehole in the subsurface
formation, said logging tool including a plurality of transmitters
and a plurality of receivers, at least one of said transmitters and
at least one of said receivers inclined to an axis of the tool,
said borehole having an axis inclined at a nonzero angle to a
normal to said layers; (b) using said electromagnetic logging tool
for obtaining a plurality of measurements with a plurality of pairs
of said transmitters and receivers; (c) using a first subset of
said plurality of measurements for determining a horizontal
conductivity associated with each of said layers; and (d) using
determined horizontal conductivities and a second subset of said
plurality of measurements for determining a vertical conductivity
associated with each of said layers.
2. The method of claim 1 wherein said plurality of transmitters
comprise x-, y- and z-transmitters and the plurality of receivers
comprise x-, y- and z-receivers.
3. The method of claim 2 wherein said first subset of measurements
consist of principal component measurements.
4. The method of claim 2 wherein determining the horizontal
conductivity and the vertical conductivity associated with each of
the plurality of layers further comprises obtaining a tool rotation
angle, formation azimuth, and an angle of inclination of said
borehole to the normal to the plurality of layers.
5. The method of claim 2 wherein said subsurface formation further
comprises a uniform formation, and the plurality of measurements
further comprises at least one measurement selected from (i) a
h.sub.xz measurement, (ii) a h.sub.xy measurement, (iii) a h.sub.zx
measurement, (iv) a h.sub.zy measurement, (v) a h.sub.yx
measurement, and, (vi) a h.sub.yz measurement.
6. The method of claim 1 wherein determining the horizontal
conductivity associated with each of said layers further comprises
applying frequency focusing to said first subset of measurements
and obtaining therefrom a first frequency focused set of
measurements.
7. The method of claim 2 wherein determining the horizontal
conductivity associated with each of said layers further comprises
applying frequency focusing to said first subset of measurements
and obtaining therefrom a second frequency focused set of
measurements.
8. The method of claim 7 wherein determining the horizontal
conductivity associated with each of said layers further comprises
determining a set of weights such that a weighted sum of the first
frequency focused set of measurements is substantially independent
of the vertical conductivity associated with each of the plurality
of layers.
9. The method of claim 8 wherein determining the vertical
conductivity associated with each of said layers further comprises
inverting the second frequency focused set of measurements using a
model including said horizontal and a vertical conductivity
associated with each of said plurality of layers.
10. The method of claim 6 wherein determining the horizontal
conductivity associated with each of said layers further comprises
determining a set of weights such that a weighted sum of the first
frequency focused set of measurements is substantially independent
of the vertical conductivity associated with each of the plurality
of layers.
11. The method of claim 10 wherein determining the vertical
conductivity associated with each of said layers further comprises
inverting the second frequency focused set of measurements using a
model including said horizontal and a vertical conductivity
associated with each of said plurality of layers.
12. The method of claim 1 wherein determining the horizontal
conductivity and the vertical conductivity associated with each of
the plurality of layers further comprises obtaining a tool rotation
angle, formation azimuth, and an angle of inclination of said
borehole to the normal to the plurality of layers.
13. The method of claim 1 further comprising repeating (c)-(d) and
iteratively updating an estimate of said non zero angle until a
difference between said measurements and a model output obtained
using said horizontal and vertical conductivities is less than a
predetermined threshold.
14. The method of claim 1 wherein determining said horizontal
conductivity associated with each of said layers further comprises
performing an inversion.
Description
BACKGROUND OF THE INVENTION
1. Field of the Invention
The invention is related generally to the field of interpretation
of measurements made by well logging instruments for the purpose of
determining the properties of earth formations. More specifically,
the invention is related to a method for determination of
anisotropic formation resistivity in a deviated wellbore using
multifrequency, multicomponent resistivity data.
2. Background of the Art
Electromagnetic induction and wave propagation logging tools are
commonly used for determination of electrical properties of
formations surrounding a borehole. These logging tools give
measurements of apparent resistivity (or conductivity) of the
formation that when properly interpreted are diagnostic of the
petrophysical properties of the formation and the fluids
therein.
The physical principles of electromagnetic induction resistivity
well logging are described, for example, in, H. G. Doll,
Introduction to Induction Logging and Application to Logging of
Wells Drilled with Oil Based Mud, Journal of Petroleum Technology,
vol. 1, p. 148, Society of Petroleum Engineers, Richardson Tex.
(1949). Many improvements and modifications to electromagnetic
induction resistivity instruments have been devised since
publication of the Doll reference, supra. Examples of such
modifications and improvements can be found, for example, in U.S.
Pat. No. 4,837,517; U.S. Pat. No. 5,157,605 issued to Chandler et
al, and U.S. Pat. No. 5,452,761 issued to Beard et al.
A limitation to the electromagnetic induction resistivity well
logging instruments known in the art is that they typically include
transmitter coils and receiver coils wound so that the magnetic
moments of these coils are substantially parallel only to the axis
of the instrument. Eddy currents are induced in the earth
formations from the magnetic field generated by the transmitter
coil, and in the induction instruments known in the art these eddy
currents tend to flow in ground loops which are substantially
perpendicular to the axis of the instrument. Voltages are then
induced in the receiver coils related to the magnitude of the eddy
currents. Certain earth formations, however, consist of thin layers
of electrically conductive materials interleaved with thin layers
of substantially non-conductive material. The response of the
typical electromagnetic induction resistivity well logging
instrument will be largely dependent on the conductivity of the
conductive layers when the layers are substantially parallel to the
flow path of the eddy currents. The substantially non-conductive
layers will contribute only a small amount to the overall response
of the instrument and therefore their presence will typically be
masked by the presence of the conductive layers. The non-conductive
layers, however, are the ones that are typically hydrocarbon
bearing and are of the most interest to the instrument user.
Interpreting a well log made using the electromagnetic induction
resistivity well logging instruments known in the art therefore may
overlook some earth formations that might be of commercial
interest.
The effect of formation anisotropy on resistivity logging
measurements has long been recognized. Kunz and Moran studied the
anisotropic effect on the response of a conventional logging device
in a borehole perpendicular to the bedding plane of t thick
anisotropic bed. Moran and Gianzero extended this work to
accommodate an arbitrary orientation of the borehole to the bedding
planes.
Rosthal (U.S. Pat. No. 5,329,448) discloses a method for
determining the horizontal and vertical conductivities from a
propagation or induction well logging device. The method assumes
the angle between the borehole axis and the normal to the bedding
plane, is known. Conductivity estimates are obtained by two
methods. The first method measures the attenuation of the amplitude
of the received signal between two receivers and derives a first
estimate of conductivity from this attenuation. The second method
measures the phase difference between the received signals at two
receivers and derives a second estimate of conductivity from this
phase shift. Two estimates are used to give the starting estimate
of a conductivity model and based on this model, an attenuation and
a phase shift for the two receivers are calculated. An iterative
scheme is then used to update the initial conductivity model until
a good match is obtained between the model output and the actual
measured attenuation and phase shift.
Hagiwara (U.S. Pat. No. 5,656,930) shows that the log response of
an induction-type logging tool can be described by an equation of
the form ##EQU1##
where
where L is the spacing between the transmitter and receiver, k is
the wavenumber of the electromagnetic wave, .mu. is the magnetic
permeability of the medium, .theta. is the deviation of the
borehole angle from the normal to the formation, .lambda. is the
anisotropy factor for the formation, .omega. is the angular
frequency of the electromagnetic wave, .sigma..sub.h is the
horizontal conductivity of the medium and .di-elect cons..sub.h is
the horizontal dielectric constant of the medium.
Eq. (3) is actually a pair of equations, one corresponding to the
real part and one corresponding to the imaginary part of the
measured signal, and has two unknowns. By making two measurements
of the measured signal, the parameters k and .beta. can be
determined. The two needed measurements can be obtained from (1) R
and X signals from induction logs, (2) phase and attenuation
measurements from induction tools, (3) phase or attenuation
measurements from induction tools with two different spacings, or
(4) resistivity measurements at two different frequencies. In the
low frequency limit, .di-elect cons. can be neglected in Eq. (3)
and from known values of .omega. and .mu., the conductivity .sigma.
can be determined from k, assuming a value of .mu. equal to the
permittivity of free space.
Wu (U.S. Pat. No. 6,092,024) recognized that the solution to eqs.
(1)-(3) may be non-unique and showed how this ambiguity in the
solution may be resolved using a plurality of measurements obtained
from multiple spacings and/or multiple frequencies.
One solution to the limitation of the induction instruments known
in the art is to include a transverse transmitter coil and a
transverse receiver coil on the induction instrument, whereby the
magnetic moments of these transverse coils is substantially
perpendicular to the axis of the instrument. Such as solution was
suggested in Tabarovsky and Epov, "Geometric and Frequency Focusing
in Exploration of Anisotropic Seams", Nauka, USSR Academy of
Science, Siberian Division, Novosibirsk, pp. 67-129 (1972).
Tabarovsky and Epov suggest various arrangements of transverse
transmitter coils and transverse receiver coils, and present
simulations of the responses of these transverse coil systems
configured as shown therein. Tabarovsky and Epov also describe a
method of substantially reducing the effect on the voltage induced
in transverse receiver coils which would be caused by eddy currents
flowing in the wellbore and invaded zone. The wellbore is typically
filled with a conductive fluid known as drilling mud. Eddy currents
that flow in the drilling mud can substantially affect the
magnitude of voltages induced in the transverse receiver coils. The
wellbore signal reduction method described by Tabarovsky and Epov
can be described as "frequency focusing", whereby induction voltage
measurements are made at more than one frequency, and the signals
induced in the transverse receiver coils are combined in a manner
so that the effects of eddy currents flowing within certain
geometries, such as the wellbore and invasion zone, can be
substantially eliminated from the final result. Tabarovsky and
Epov, however, do not suggest any configuration of signal
processing circuitry which could perform the frequency focusing
method suggested in their paper.
Strack et al. (U.S. Pat. No. 6,147,496) describe a multicomponent
logging tool comprising a pair of 3-component transmitters and a
pair of 3-component receivers. Using measurements made at two
different frequencies, a combined signal is generated having a
reduced dependency on the electrical conductivity in the wellbore
region. U.S. Pat. No. 5,781,436 to Forgang et al, the contents of
which are fully incorporated herein by reference, discloses a
suitable hardware configuration for multifrequency, multicomponent
induction logging.
U.S. Pat. No. 5,999,883 issued to Gupta et al, (the "Gupta
patent"), the contents of which are fully incorporated here by
reference, discloses a method for determination of an initial
estimate of the horizontal and vertical conductivity of anisotropic
earth formations. Electromagnetic induction signals induced by
induction transmitters oriented along three mutually orthogonal
axes are measured at a single frequency. One of the mutually
orthogonal axes is substantially parallel to a logging instrument
axis. The electromagnetic induction signals are measured using
first receivers each having a magnetic moment parallel to one of
the orthogonal axes and using second receivers each having a
magnetic moment perpendicular to a one of the orthogonal axes which
is also perpendicular to the instrument axis. A relative angle of
rotation of the perpendicular one of the orthogonal axes is
calculated from the receiver signals measured perpendicular to the
instrument axis. An intermediate measurement tensor is calculated
by rotating magnitudes of the receiver signals through a negative
of the angle of rotation. A relative angle of inclination of one of
the orthogonal axes that is parallel to the axis of the instrument
is calculated, from the rotated magnitudes, with respect to a
direction of the vertical conductivity. The rotated magnitudes are
rotated through a negative of the angle of inclination. Horizontal
conductivity is calculated from the magnitudes of the receiver
signals after the second step of rotation. An anisotropy parameter
is calculated from the receiver signal magnitudes after the second
step of rotation. Vertical conductivity is calculated from the
horizontal conductivity and the anisotropy parameter. One drawback
in the teachings of Gupta et al is that the step of determination
of the relative angle of inclination of the required measurements
of three components of data with substantially identical
transmitter-receiver spacings. Because of limitations on the
physical size of the tools, this condition is difficult to obtain;
consequently the estimates of resistivities are susceptible to
error. In addition, due to the highly nonlinear character of the
response of multicomponent tools, such inversion methods are time
consuming at a single frequency and even more so at multiple
frequencies.
Analysis of the prior art leads to the conclusion that known
methods of determining anisotropic resistivities in real time
require very low frequencies; as a consequence of the low
frequencies, the signal-to-noise ratio in prior art methods is
quite low.
Co-pending U.S. patent application Ser. No. 09/825,104, (referred
to hereafter as the '104 application) filed on Apr. 3, 2001 teaches
a computationally fast method of determination of horizontal and
vertical conductivities of subsurface formations using a
combination of data acquired with a transverse induction logging
tool such as the 3DEX.SM. tool and data acquired with a
conventional high definition induction logging tool (HDIL). The
data are acquired in a vertical borehole. The HDIL data are used to
determine horizontal resistivities that are used in an isotropic
model to obtain expected values of the transverse components of the
3DEX.SM.. Differences between the model output and the acquired
3DEX.SM. data are indicative of anisotropy and this difference is
used to derive an anisotropy factor. The method described therein
has difficulties in deviated boreholes as the HDIL measurements
used to derive the isotropic model are responsive to both
horizontal and vertical resistivity.
There is a need for a fast and robust method of determination of
anisotropic resistivity. Such a method should preferably be able to
use high frequency measurements that are known to have better
signal-to-noise ratio than low frequency methods. The present
invention satisfies this need.
SUMMARY OF THE INVENTION
A method of logging subsurface formations using data acquired with
a transverse induction logging tool, the formation having a
horizontal conductivity and a vertical conductivity, by obtaining a
plurality of frequencies measurements indicative of vertical and
horizontal conductivities in a tool referenced coordinate system.
The data are transformed to a subsurface formation coordinate
system. Multifrequency focusing is applied to the measurements at a
plurality of frequencies. Horizontal formation conductivities are
determined from a subset of the focused conductivity measurements.
Vertical formation conductivities are determined from the focused
conductivity measurements associated with the subsurface formation
and the horizontal conductivities.
In a preferred embodiment of the invention, a transformation
independent of the formation azimuth may be used to determine the
conductivity of the transversely anisotropic formation.
BRIEF DESCRIPTION OF THE FIGURES
FIG. 1(Prior art) shows an induction instrument disposed in a
wellbore penetrating earth formations.
FIG. 2 shows the arrangement of transmitter and receiver coils in a
preferred embodiment of the present invention marketed under the
name 3DEX.SM..
FIG. 3 shows an earth model example used in the present
invention.
FIG. 4 is a flow chart illustrating steps comprising the present
invention.
DETAILED DESCRIPTION OF THE INVENTION
FIG. 1 (prior art) shows an induction well logging instrument 10
disposed in a wellbore 2 penetrating earth formations. The earth
formations are shown generally at 6, 8, 12 and 14. The instrument
10 is typically lowered into the wellbore 2 at one end of an
armored electrical cable 22, by means of a winch 28 or similar
device known in the art. An induction well logging instrument which
will generate appropriate types of signals for performing the
process of the present invention is described, for example, in U.S.
Pat. No. 5,452,761 issued to Beard et al. The prior art induction
logging tool includes a transmitter coil and a plurality of
receiver coils 18A-18F. The coils in the prior art device are
oriented with the coil axes parallel to the axis of the tool and to
the wellbore.
Turning now to FIG. 2, the configuration of transmitter and
receiver coils in a preferred embodiment of the 3DExplorer.TM.
induction logging instrument of Baker Hughes is disclosed. Such a
logging instrument is an example of a transverse induction logging
tool. Three orthogonal transmitters 101, 103 and 105 that are
referred to as the T.sub.x, T.sub.z, and T.sub.y transmitters are
shown (the z-axis is the longitudinal axis of the tool).
Corresponding to the transmitters 101, 103 and 105 are receivers
107, 109 and 111, referred to as the R.sub.x, R.sub.z, and R.sub.y
receivers, for measuring the corresponding components (H.sub.xx,
H.sub.yy, H.sub.zz) of induced signals. In addition,
cross-components are also measured. These are denoted by H.sub.xy,
H.sub.xz etc.
FIG. 3 is a schematic illustration of the model used in the present
invention. The subsurface of the earth is characterized by a
plurality of layers 201a, 201b, . . . , 201i. The layers have
thicknesses denoted by h.sub.1, h.sub.2, . . . h.sub.i. The
horizontal and vertical resistivities in the layers are denoted by
R.sub.h1, R.sub.h2, . . . R.sub.hi and R.sub.v1, R.sub.v2, . . .
R.sub.vi respectively. Equivalently, the model may be defined in
terms of conductivities (reciprocal of resistivity). The borehole
is indicated by 202 and associated with each of the layers are
invaded zones in the vicinity of the borehole wherein borehole
fluid has invaded the formation and altered is properties so that
the electrical properties are not the same as in the uninvaded
portion of the formation. The invaded zones have lengths L.sub.x01,
L.sub.x02, . . . L.sub.x0i extending away from the borehole. The
resistivities in the invaded zones are altered to values R.sub.x01,
R.sub.x02, . . . R.sub.x0i. In the embodiment of the invention
discussed here, the invaded zones are assumed to be isotropic while
an alternate embodiment of the invention includes invaded zones
that are anisotropic, i.e., they have different horizontal and
vertical resistivities. It should further be noted that the
discussion of the invention herein may be made in terms of
resistivities or conductivities (the reciprocal of resistivity).
The z-component of the 3DEX.SM. tool is oriented along the borehole
axis and makes an angle .theta.(not shown) with the normal to the
bedding plane. The x-component of the tool makes an angle .phi.
with the "up" direction.
In the '104 application to Tabarovsky, et al. multifrequency,
multicomponent induction data are obtained using, for example, the
3DEX.TM. tool, and a multifrequency focusing is applied to these
data. As disclosed in U.S. Pat. No. 5,703,773 to Tabarovsky et al.,
the contents of which are fully incorporated herein by reference,
the response at multiple frequencies may be approximated by a
Taylor series expansion of the form: ##EQU2##
In a preferred embodiment of the invention of the '104 application,
the number m of frequencies .omega. is ten. In eq.(4), n is the
number of terms in the Taylor series expansion. This can be any
number less than or equal to m. The coefficient s.sub.3/2 of the
.omega..sup.3/2 term (.omega. being the square of k, the wave
number) is generated by the primary field and is relatively
unaffected by any inhomogeneities in the medium surround the
logging instrument, i.e., it is responsive primarily to the
formation parameters and not to the borehole and invasion zone. In
fact, the coefficient s.sub.3/2 of the .omega..sup.3/2 term is
responsive to the formation parameters as though there were no
borehole in the formation. Specifically, these are applied to the
H.sub.xx and H.sub.yy components. Those versed in the art would
recognize that in a vertical borehole, the H.sub.xx and H.sub.yy
would be the same, with both being indicative of the vertical
conductivity of the formation. In one embodiment of the invention,
the sum of the H.sub.xx and H.sub.yy is used so as to improve the
signal to noise ratio (SNR). This multifrequency focused
measurement is equivalent to the zero frequency value. As would be
known to those versed in the art, the zero frequency value may also
be obtained by other methods, such as by focusing using focusing
electrodes in a suitable device.
Along with the 3DEX.TM., the method of the '104 application also
uses data from a prior art High Definition Induction Logging (HDIL)
tool having transmitter and receiver coils aligned along the axis
of the tool. These data are inverted using a method such as that
taught by Tabarovsky et al, or by U.S. Pat. No. 5,884,227 to
Rabinovich et al., the contents of which are fully incorporated
herein by reference, to give an isotropic model of the subsurface
formation. Instead of, or in addition to the inversion methods, a
focusing method may also be used to derive the initial model. Such
focusing methods would be known to those versed in the art and are
not discussed further here. As discussed above, the HDIL tool is
responsive primarily to the horizontal conductivity of the earth
formations when run in a borehole that is substantially orthogonal
to the bedding planes. The inversion methods taught by Tabarovsky
et al and by Rabinovich et al are computationally fast and may be
implemented in real time. These inversions give an isotropic model
of the horizontal conductivities (or resistivities)
Using the isotropic model derived, a forward modeling is used in
the '104 application to calculate a synthetic response of the
3DEX.TM. tool at a plurality of frequencies. A suitable forward
modeling program for the purpose is disclosed in Tabarovsky and
Epov "Alternating Electromagnetic Field in an Anisotropic Layered
Medium" Geol. Geoph., No. 1, pp. 101-109. (1977). Multifrequency
focusing is then applied to these synthetic data. In a preferred
embodiment of the invention of the '104 application, the method
taught by Tabarovsky is used for the purpose.
In the absence of anisotropy, the output from model output should
be identical to the multifrequency focused measurements. Denoting
by .sigma..sub.iso the multifrequency focused transverse component
synthetic data from and by .sigma..sub.meas the multifrequency
focused field data from, the anisotropy factor .lambda. is then
calculated in the '104 application.
The H.sub.xx for an anisotropic medium is given by ##EQU3##
For a three-coil subarray, ##EQU4##
Upon introducing the apparent conductivity for H.sub.xx this gives
##EQU5##
which gives the result ##EQU6##
where .sigma..sub.t is the conductivity obtained from the HDIL
data, i.e., the horizontal conductivity. The vertical conductivity
is obtained by dividing .sigma..sub.t by the anisotropy factor from
eq. (5). An important aspect of the '104 application is that in a
vertical borehole, the measurements made by a HDIL tool depend only
on the horizontal conductivities and not on the vertical
resistivities. The method of the invention disclosed there is to
obtain an isotropic model from the HDIL data, use the isotropic
model to predict measurements made on other components and to use a
difference between the predicted and actual measurements to obtain
the vertical conductivity.
In a similar manner, the method of the present invention can be
viewed as finding a combination of 3DEX.SM. measurements (called
modes of the induction measurements) that are responsive only to
the horizontal conductivity, deriving a model of horizontal
conductivity from this combination of measurements, predicting
values of other components of 3DEX.SM. measurements and using a
difference between these predicted measurements and the actual
measurements to determine a vertical conductivity. In a particular
embodiment of the present invention, the desired combination
includes only the principal component measurements, i.e., upon
H.sub.xx, H.sub.yy, and H.sub.zz. The flow chart of the method of
the present invention is shown in FIG. 4
The method of the present invention starts with an estimate of the
dip and azimuth of the formation relative to the borehole axis 300.
These angles are defined below. In addition, a sensor on the
logging tool also provides another angle measurement called the
"toolface angle" that is also used in the analysis of the data.
Multifrequency focused data are derived from multifrequency
measurements 301. The data are transformed 303 as discussed below
to give measurements that are indicative only of horizontal
conductivity. These data are inverted 305 to give a model of the
horizontal conductivity of the data. These estimates of horizontal
conductivity are used in an isotropic model as estimates of the
vertical conductivity 307. The measured data are then inverted
using this initial estimate of vertical conductivities 309. A check
is made of the goodness of fit (difference between a model output
based on the inverted model and the actual measurements) 311. If
the difference is below a predetermined threshold, then the model
is accepted. If the difference is excessive, a iterative procedure
is carried out with an updated angle estimate 313 until the result
is acceptable. Any suitable iterative procedure may be used such as
that based on a gradient method or a method of steepest descent.
Such iterative methods would be known to those versed in the art
and are not discussed further.
At this point we develop the principal component structure for
measuring formation anisotropy in bedding planes when the borehole
is not normal (perpendicular) to the bedding plane. For simplifying
the notation, and to avoid confusion, the x-, y- and z-components
in the tool coordinates are called hereafter the (1, 2, 3)
coordinate system. The x-, y- and z-components in the earth
coordinate system will be referred to as such.
In the tool coordinate system, the matrix of magnetic components,
H.sub.T, may be represented in the following form: ##EQU7##
For layered formations, the matrix, H.sub.T, is symmetric. We
measure three diagonal elements, h.sub.11, h.sub.22, and h.sub.33.
The non-diagonal elements are not needed in the present
invention.
In the earth coordinate system, {x, y, z}, associated with the
plane formation boundaries (z-axis is perpendicular to the
boundaries and directed downwards) the magnetic matrix may be
presented as follows: ##EQU8##
The formation resistivity is described as a tensor, .rho.. In the
coordinate system associated with a formation, the resistivity
tensor has only diagonal elements in the absence of azimuthal
anisotropy: ##EQU9## .rho..sub.t =.rho..sub.xx =.rho..sub.yy,
.rho..sub.n =.rho..sub.zz
The "tool coordinate" system (1-, 2-, 3-) can be obtained from the
"formation coordinate" system (x-, y-, z-) as a result of two
sequential rotations: (1) Rotation about the axis "2" by the angle
.theta., such that the axis "3" in a new position (let us call it
"3'") becomes parallel to the axis z of the "tool" system; (2)
Rotation about the axis "3'" by the angle .PHI., such that the new
axis "1" (let us call it "1'") becomes parallel to the axis x of
the tool system.
In the present invention, an iterative procedure as shown in FIG. 4
is used for .theta. and .phi.. .theta. is the relative inclination
of the borehole axis to the normal to the bedding while .phi. is
the azimuth. An initial estimate for .theta. and .phi. may be
determined from borehole surveys or from resistivity imaging
devices and from knowledge of the toolface angle.
The first rotation is described using matrices .theta. and
.theta..sup.T : ##EQU10##
Here, C.sub..theta. =cos .theta., S.sub..theta. =sin .theta.
The second rotation is described using matrices .PHI. and
.PHI..sup.T : ##EQU11##
Here, C.sub..PHI. =cos .PHI., S.sub..PHI. =sin .PHI.
Matrices H.sub.M (the formation coordinate system) and H.sub.T (the
tool coordinate system) are related as follows:
It is worth noting that the matrix H.sub.M contains zero
elements:
This is true for multifrequency focused measurements as described
below. It is also important to note that the following three
components of the matrix H.sub.M depend only on the horizontal
resistivity.
Two remaining elements depend on both horizontal and vertical
resistivities.
Taking into account Equations (11), (12), (14) and (15), we can
re-write Equation (13) as follows: ##EQU12##
The following expanded calculations are performed in order to
present Equation (18) in a form more convenient for analysis.
##EQU13##
The components of A.sub.3 are given as
[a.sub.11.sup.(3) =C.sub..theta..sup.2 C.sub..PHI. h.sub.xx
-2C.sub..theta. S.sub..theta. C.sub..PHI. h.sub.xz -S.sub..theta.
S.sub..PHI. h.sub.yz +S.sub..theta..sup.2 C.sub..PHI. h.sub.zz
](*
a.sub.33.sup.(3) =S.sub..theta..sup.2 h.sub.xx +C.sub..theta.
S.sub..theta. h.sub.xz +C.sub..theta. S.sub..theta. h.sub.xz
+C.sub..theta..sup.2 h.sub.zz
Taking into account all the above calculations, we can represent
Equation (18) in the following form: ##EQU14##
The method of the present invention involves defining a linear
combination of the measurements that are responsive substantially
to the horizontal conductivity and not responsive to the vertical
conductivity. In a preferred embodiment of the invention, the
linear combination is that of measurements h.sub.11, h.sub.22, and
h.sub.33 (i.e., the principal components only), although in
alternate embodiments of the invention, a linear combination of any
of the measurements may be used. The example given below is that of
the preferred embodiment.
Let us consider expressions for the measured principal components,
h.sub.11, h.sub.22, and h.sub.33 : ##EQU15##
More detailed representation yields:
Expressions for each component, h.sub.11, h.sub.22, and h.sub.33,
contain two types of functions: some depending only on .rho..sub.t,
and some others depending on both, .rho..sub.t and .rho..sub.n.
Equations (13)-(15) may be represented in the following form:
##EQU16##
A linear combination of Equations (23) is defined in the form:
Detailed consideration of Equation (25) yields:
The method of the present invention involves defining the
coefficients, .alpha. and .beta., in such a way that the resulting
linear combination, h, does not depend on the vertical resistivity.
To achieve that, we need to null the following part of the
expression for h:
Imposing the following conditions satisfies equation (26):
##EQU17##
Let us calculate the coefficients, .alpha. and .beta.. The second
Equation in (27) yields: ##EQU18##
After substitution of Equation (28) in the first Equation of (27),
we obtain: ##EQU19##
To obtain the coefficient, .beta., let us substitute Equation (29)
in Equation (28): ##EQU20##
Finally, ##EQU21##
It is convenient to normalize coefficients, .alpha. and .beta.. Let
us introduce a normalization factor, .kappa..
Equation (25) may be presented in the form:
Here,
Calculations yield: ##EQU22## ##EQU23##
Consequently, ##EQU24##
Finally, we obtain: ##EQU25##
Here,
The coefficient, .kappa., degenerates under the following
conditions:
Using the derivation given above, for an estimated value of .theta.
and .phi., the conductivities may be derived. A difference between
the model output and the measured values may then be used in the
iterative procedure described with respect to FIG. 4.
The derivation above has been done for a single frequency data.
Multifrequency Focused (MFF) data is a linear combination of single
frequency measurements so that the derivation given above is
equally applicable to MFF data. It can be proven that the three
principle 3DEX.TM. measurements, MFF (multi-frequency focusing)
processed, may be expressed in the following form: ##EQU26##
The matrix coefficients of Equation 39 depend on .theta..sub.r,
.phi..sub.r, and three trajectory measurements: deviation, azimuth
and rotation.
The components of the vector in the right hand side of Equation 39
represent all non-zero field components generated by three
orthogonal induction transmitters in the coordinate system
associated with the formation. Only two of them depend on vertical
resistivity: h.sub.xx and h.sub.yy. This allows us to build a
linear combination of measurements, h.sub.11, h.sub.22, and
h.sub.33 in such a way that the resulting transformation depends
only on h.sub.zz and h.sub.xz, or, in other words, only on
horizontal resistivity. Let T be the transformation with
coefficients .alpha., .beta. and .gamma.:
The coefficients .alpha., .beta. and .gamma. must satisfy the
following system of equations:
From the above discussion it follows that a transformation may be
developed that is independent of the formation azimuth. The
formation azimuth-independent transformation may be expressed
as:
where .theta. is the dip of the formation and T.sub.0 is the linear
transformation to separate modes. With this transformation and the
above series of equations we may determine the conductivity of the
transversely anisotropic formation.
It is to be noted, however, that when the earth formation is
uniform (i.e., there are no formation boundaries within the region
of investigation of the tool), it is not possible to satisfy eq.
(40). It is necessary then to have a measurement of at least one
cross-component.
The present invention has been discussed above with respect to
measurements made by a transverse induction-logging tool conveyed
on a wireline. This is not intended to be a limitation and the
method is equally applicable to measurements made using a
comparable tool conveyed on a measurement-while-drilling (MWD)
assembly or on coiled tubing.
While the foregoing disclosure is directed to the preferred
embodiments of the invention, various modifications will be
apparent to those skilled in the art. It is intended that all
variations within the scope and spirit of the appended claims be
embraced by the foregoing disclosure.
* * * * *